Introduction: This paper represents the numerical simulation of blood flow in the circle of Willis (CoW). Circle of Willis is responsible for the oxygenated blood distribution into the cerebral mass. To investigate the blood behavior, two Newtonian and non-Newtonian viscosity models were considered and the results were compared under steady state conditions. Methods: Methodologically, the arterial geometry was obtained using 3D magnetic resonance angiography (MRA) data. The blood flow through the cerebral vasculature was considered to be steady and laminar, and the Galerkin’s finite element method was applied to solve the systems of non-linear Navier-Stokes equations. Results: Flow patterns including flow rates and shear rates were obtained through the simulation. The minimal magnitude of shear rates was much greater than 100 s-1 through the larger arteries; thus, the non-Newtonian blood viscosity tended to approach the constant limit of infinite shear viscosity through the CoW. So, in larger arteries the non-Newtonian nature of blood was less dominant and it would be treated as a Newtonian fluid. The only exception was the anterior communicating artery (ACoA) in which the blood flow showed different behavior for the Newtonian and non-Newtonian cases. Conclusion By comparing the results it was concluded that the Newtonian viscosity assumption of blood flow through the healthy, complete circle of Willis under the normal and steady conditions would be acceptably accurate.